I have 18 updates appearing for Windows 7 x64 but when I try to install any of them I get the error 800706BE.

The Microsoft help that appears gives several solutions:

Fix It button. This downloads MicrosoftFixit50123.msi (963KB). After running it displays the message "This Microsoft Fix It has been processed".

System Update Readiness Tool. This is actually in response to error 800B0100 which isn't the one I'm seeing. You download the file here (362 MB). After running it says "Do you want to install the following Windows software update: Hotfix for Windows (KB947821)"